Weyl's formula describes the characters of the irreducible representations of compact connected Lie groups. Atiyah and Bott pointed out a long time ago that there are close connections between the character formula and K-theory. I shall reexamine those connections in these lectures, partly to illustrate some of the basic features of K-theory, and partly to prepare for the case of noncompact groups, where similar connections ought to link the Baum-Connes conjecture to geometric representation theory. I shall begin with an introductory account of the character formula in lecture one, and then discuss basic K-theory in lecture 2 before venturing toward more specialized topics.